Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $t \neq 0$. $q = \dfrac{6(2t - 1)}{3t} \div \dfrac{10t - 5}{7t} $
Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{6(2t - 1)}{3t} \times \dfrac{7t}{10t - 5} $ When multiplying fractions, we multiply the numerators and the denominators. $q = \dfrac{ 6(2t - 1) \times 7t } { 3t \times (10t - 5) } $ $ q = \dfrac {7t \times 6(2t - 1)} {3t \times 5(2t - 1)} $ $ q = \dfrac{42t(2t - 1)}{15t(2t - 1)} $ We can cancel the $2t - 1$ so long as $2t - 1 \neq 0$ Therefore $t \neq \dfrac{1}{2}$ $q = \dfrac{42t \cancel{(2t - 1})}{15t \cancel{(2t - 1)}} = \dfrac{42t}{15t} = \dfrac{14}{5} $